Solve 25x^2-46x+16

$$25 x ^ { 2 } - 46 x + 16$$

Factor

$$25\left(x-\frac{23-\sqrt{129}}{25}\right)\left(x-\frac{\sqrt{129}+23}{25}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$25x^{2}-46x+16=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-46\right)±\sqrt{\left(-46\right)^{2}-4\times 25\times 16}}{2\times 25}$$

Square -46.

$$x=\frac{-\left(-46\right)±\sqrt{2116-4\times 25\times 16}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-46\right)±\sqrt{2116-100\times 16}}{2\times 25}$$

Multiply -100 times 16.

$$x=\frac{-\left(-46\right)±\sqrt{2116-1600}}{2\times 25}$$

Add 2116 to -1600.

$$x=\frac{-\left(-46\right)±\sqrt{516}}{2\times 25}$$

Take the square root of 516.

$$x=\frac{-\left(-46\right)±2\sqrt{129}}{2\times 25}$$

The opposite of -46 is 46.

$$x=\frac{46±2\sqrt{129}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{46±2\sqrt{129}}{50}$$

Now solve the equation x=\frac{46±2\sqrt{129}}{50} when ± is plus. Add 46 to 2\sqrt{129}.

$$x=\frac{2\sqrt{129}+46}{50}$$

Divide 46+2\sqrt{129} by 50.

$$x=\frac{\sqrt{129}+23}{25}$$

Now solve the equation x=\frac{46±2\sqrt{129}}{50} when ± is minus. Subtract 2\sqrt{129} from 46.

$$x=\frac{46-2\sqrt{129}}{50}$$

Divide 46-2\sqrt{129} by 50.

$$x=\frac{23-\sqrt{129}}{25}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{23+\sqrt{129}}{25} for x_{1} and \frac{23-\sqrt{129}}{25} for x_{2}.

$$25x^{2}-46x+16=25\left(x-\frac{\sqrt{129}+23}{25}\right)\left(x-\frac{23-\sqrt{129}}{25}\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 25

$$x ^ 2 -\frac{46}{25}x +\frac{16}{25} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{46}{25} rs = \frac{16}{25}$$

Two numbers r and s sum up to \frac{46}{25} exactly when the average of the two numbers is \frac{1}{2}*\frac{46}{25} = \frac{23}{25}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{23}{25} - u s = \frac{23}{25} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{16}{25}

$$(\frac{23}{25} - u) (\frac{23}{25} + u) = \frac{16}{25}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{529}{625} - u^2 = \frac{16}{25}$$

Simplify the expression by subtracting \frac{529}{625} on both sides

$$-u^2 = \frac{16}{25}-\frac{529}{625} = -\frac{129}{625}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{129}{625} u = \pm\sqrt{\frac{129}{625}} = \pm \frac{\sqrt{129}}{25}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{23}{25} - \frac{\sqrt{129}}{25} = 0.466 s = \frac{23}{25} + \frac{\sqrt{129}}{25} = 1.374$$

Evaluate

$$25x^{2}-46x+16$$